Normalized defining polynomial
\( x^{13} + 7x - 7 \)
Invariants
Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(5056053485497795123654045\) \(\medspace = 5\cdot 7^{12}\cdot 23\cdot 929\cdot 21557\cdot 158611\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(79.49\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $5^{1/2}7^{12/13}23^{1/2}929^{1/2}21557^{1/2}158611^{1/2}\approx 115188225.55455989$ | ||
Ramified primes: | \(5\), \(7\), \(23\), \(929\), \(21557\), \(158611\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{365287809730045}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $a-1$, $5a^{12}+7a^{11}+4a^{10}-4a^{9}-11a^{8}-12a^{7}-3a^{6}+11a^{5}+20a^{4}+16a^{3}-4a^{2}-25a+1$, $4a^{12}+a^{11}-a^{10}+a^{9}-2a^{8}-5a^{7}+3a^{6}+5a^{5}-6a^{4}-4a^{3}+3a^{2}-14a+15$, $9a^{12}+3a^{11}-21a^{10}+45a^{9}-76a^{8}+113a^{7}-155a^{6}+198a^{5}-237a^{4}+264a^{3}-270a^{2}+246a-113$, $52a^{12}+38a^{11}-20a^{10}-93a^{9}-92a^{8}-10a^{7}+75a^{6}+144a^{5}+149a^{4}-22a^{3}-242a^{2}-246a+272$, $65a^{12}+98a^{11}+120a^{10}+128a^{9}+130a^{8}+109a^{7}+62a^{6}-10a^{5}-76a^{4}-130a^{3}-189a^{2}-248a+169$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 9863204.921 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 9863204.921 \cdot 2}{2\cdot\sqrt{5056053485497795123654045}}\cr\approx \mathstrut & 0.5397857712 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 6227020800 |
The 101 conjugacy class representatives for $S_{13}$ |
Character table for $S_{13}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 26 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.5.0.1}{5} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | R | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.13.0.1}{13} }$ | ${\href{/padicField/31.13.0.1}{13} }$ | ${\href{/padicField/37.13.0.1}{13} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.11.0.1 | $x^{11} + 3 x + 3$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
\(7\) | 7.13.12.1 | $x^{13} + 7$ | $13$ | $1$ | $12$ | $F_{13}$ | $[\ ]_{13}^{12}$ |
\(23\) | $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.10.0.1 | $x^{10} + 17 x^{5} + 5 x^{4} + 15 x^{3} + 6 x^{2} + x + 5$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(929\) | $\Q_{929}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(21557\) | $\Q_{21557}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(158611\) | $\Q_{158611}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{158611}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |